Paper:

Explicit, almost optimal, epsilon-balanced codes

The question of finding an epsilon-biased set with close to optimal support size, or, equivalently, finding an explicit binary code with distance $\frac{1-\epsilon}{2}$ and rate close to the Gilbert-Varshamov bound, attracted a lot of attention in recent decades. In this paper we solve the problem almost optimally and show an explicit $\epsilon$-biased set over $k$ bits with support size $O(\frac{k}{\epsilon^{2+o(1)}})$. This improves upon all previous explicit constructions which were in the order of $\frac{k^2}{\epsilon^2}$, $\frac{k}{\epsilon^3}$ or $\frac{k^{5/4}}{\epsilon^{5/2}}$. The result is close to the Gilbert-Varshamov bound which is $O(\frac{k}{\epsilon^2})$ and the lower bound which is $\Omega(\frac{k}{\epsilon^2 \log \frac{1}{\epsilon}})$.

The main technical tool we use is bias amplification with the $s$-wide replacement product. The sum of two independent samples from an $\epsilon$-biased set is $\epsilon^2$ biased. Rozenman and Wigderson showed how to amplify the bias more economically by choosing two samples with an expander. Based on that they suggested a recursive construction that achieves sample size $O(\frac{k}{\epsilon^4})$. We show that amplification with a long random walk over the $s$-wide replacement product reduces the bias almost optimally.